Bifurcation analysis and new waveforms to the first fractional WBBM equation

This research focuses on bifurcation analysis and new waveforms for the first fractional 3D Wazwaz–Benjamin–Bona–Mahony (WBBM) structure, which arises in shallow water waves. The linear stability technique is also employed to assess the stability of the mentioned model. The suggested equation’s dynamical system is obtained by applying the Galilean transformation to achieve our goal. Subsequently, bifurcation, chaos, and sensitivity analysis of the mentioned model are conducted by applying the principles of the planar dynamical system. We obtain periodic, quasi-periodic, and chaotic behaviors of the mentioned model. Furthermore, we introduce and delve into diverse solitary wave solutions, encompassing bright soliton, dark soliton, kink wave, periodic waves, and anti-kink waves. These solutions are visually presented through simulations, highlighting their distinct characteristics and existence. The results highlight the effectiveness, brevity, and efficiency of the employed integration methods. They also suggest their applicability to delving into more intricate nonlinear models emerging in modern science and engineering scenarios. The novelty of this research lies in its detailed analysis of the governing model, which provides insights into its complex dynamics and varied wave structures. This study also advances the understanding of nonlinear wave properties in shallow water by combining bifurcation analysis, chaotic behavior, waveform characteristics, and stability assessments.

predictive models and decision-making processes in related fields, which significantly contribute to advancing knowledge of nonlinear wave dynamics and its practical applications.
By applying the techniques proposed for Eq.(7), this study provided novel findings that had not formerly been reported.
The structure of the existent investigation is outlined as follows: "Conformable derivative and its features" section contains the conformable derivative and its features.In "Suggested model" and "Ordinary differential form of the suggested model" sections cover the descriptions and the ordinary differential form of the governing model, respectively.Bifurcation analysis is detailed in "Bifurcation analysis" section.In "Chaotic behaviors" section, the chaotic natures of the stated model are presented.In "Sensitivity analysis" section encompasses the sensitivity analysis of the suggested model.In "Bright and dark solitons of the WBBM model" section provides dark and bright solitons of the mentioned model.The discussion of the results is contained in "Discussion of results" section.Stability analysis is given in "Stability analysis" section.In "Novelty of the outcomes" section discusses the novelty of the outcomes.Lastly, "Conclusion" section encapsulates the findings drawn from this study.

Suggested model
The BBM equation was first observed in 1972 as a development of the KdV equation for shallow waves of water in a homogeneous system.The BBM equation is frequently used as a variant of the KdV equation for describing shallow water waves.It is applicable not only to surface waves in water but also to drift waves and Rossby waves in plasma within spinning environments.The BBM nonlinear model 52 is given in the following manner: and the KdV equation is given by Both the BBM and KdV equations serve as foundational tools for comprehending various wave phenomena.They serve as essential tools for analyzing surface waves in water bodies, hydromagnetic waves in plasma, acoustical and gravitational waves in compressible fluids, acoustical waves in harmonic crystals, and long waves in nonlinear dispersive processes, among other applications.A new equation called the Wazwaz-Benjamin-Bona-Mahony (WBBM) equation 53 was derived by Wazwaz in 2017 from a 3-dimensional modified BMM equation, which is expressed as Wazwaz reformulated these newly introduced equations.This article will focus on the first fractional 3D WBBM equation, as presented in 54 , which is given in the following manner: where q(t, x, y, z) is a wave function with 4 free components t, x, y, and z.D α t , D α x , D α y , and D α z correspond to the fractional derivatives of order α w.r. to t, x, y, and z, sequentially whilst 0 < α ≤ 1, t ≥ 0.
www.nature.com/scientificreports/with real constants a 1 , a 2 , a 3 , and a 4 .Applying Eq. ( 8) in Eq. ( 7) gives the next outcome as Integrating Eq. ( 9) w. r. to θ , one obtains with integration constant a 5 .For simplicity a 5 , by setting a 5 to 0, one reaches the next ordinary differential form as

Bifurcation analysis
This segment introduces the bifurcation and phase portraits of the next planner dynamical system.Nonlinear partial differential models can be analyzed qualitatively using this dynamic system method.The orbits of this system can manifest as points, simple closed curves, or other isomorphic curves.These varied orbits correspond to the solutions of Eq. ( 7) with distinct physical characteristics.Consider dQ dθ = P , then the planner dynamical form of Eq. ( 7) with Hamiltonian function can be written as where l = a 2 a 1 a 3 a 4 , m = a 1 −a 4 a 1 a 3 a 4 , and h is the Hamiltonian constant.Let Q(θ) be a solution to Equation Eq. ( 12) with the conditions where u 1 and u 2 are constants.In the case where u 1 = u 2 , Q(θ) signifies a homoclinic orbit, leading to Q(θ) emerg- ing a solitary wave solution of Eq. (11).Conversely, if u 1 = u 2 , then Q(θ) corresponds to a heteroclinic orbit.Specifically, for u 1 > u 2 , Q(θ) takes the form of a kink wave solution, while for u 1 < u 2 , it becomes an anti-kink wave solution.Another scenario arises when Eq. ( 12) exhibits a closed phase portrait, resulting in Eq. ( 11) having a periodic solution.It is noted that a phase portrait represents a collection of orbits in a phase plane.
Case 4: l < 0, m < 0 By selecting a parameter set such that a 1 = 2, a 3 = a 4 = 1, a 2 = −1 , there is only one equilibrium point (0, 0), is identified, as presented in Fig. 1d.In this scenario, (0, 0) represents a saddle point.We see that there is no closed trajectory for the system Eq.(12).

Chaotic behaviors
This section examines the chaotic features of the resulting dynamical system by considering perturbed terms.This analysis is conducted through the examination of 2D and 3D phase portraits.To initiate this investigation, let us consider the dynamical system: where σ cos(ωθ) represents the perturbed term, σ corresponds to the amplitude, and ω is the frequency of the system.In this section, we explore how the perturbation's intensity and frequency impact the system Eq.( 15).Keeping the main parameters fixed ( a 1 = a 2 = 1, a 3 = 1 4 , a 4 = 2 ), we obtain quasiperiodic and chaotic behaviors for diverse strengths and frequencies in Figs. 2, 3, and 4. Figure 2 signifies the state of Eq. ( 15) when σ = 0 .We display the trajectory's status depending on the perturbation strength and frequency.Figure 2 displays the periodic nature of the system Eq.( 15) in time series, 2D-, and 3D phase projections.The results of Fig. 3 show that the dynamic system changes from a period to a quasi-period with a small variation in strength and frequency ( σ increases to 0.3 and ω = 0.2 ).In Fig. 4, with increased frequency and strength ( σ increases to 2.9 and ω = 3.9 ), the system undergoes violent disturbances, transitioning into a chaotic state.

Sensitivity analysis
This portion investigates the impact of initial values on the perturbed system Eq.( 15) across a range of strengths and frequencies, maintaining constant parameter values ( a 1 = a 2 = 1, a 3 = 1 4 , a 4 = 2 ).The results, depicted in Fig. 5, showcase a red curve representing a time series plot with initial values (Q(0), P(0)) = (1, 0.1) and a blue curve with (Q(0), P(0)) = (1, 0.2) .In Fig. 5a, it is evident that the periodic nature of the outcome is determined by the initial value of the perturbed system (σ = 0).Figure 5b illustrates that with a small perturbation strength ( σ = 0.3 ), the two-time series diagrams exhibit only small changes, indicating low sensitivity to the initial con- dition.Conversely, when the perturbation strength increases ( σ = 2.9 ), Fig. 5c reveals major changes between time series diagrams, signifying heightened sensitivity to changes in the initial value.

Bright and dark solitons of the WBBM model
This section examines dark and bright solitons obtained from the mentioned model employing the planner dynamical system technique.
Case 1: l < 0, and m > 0 For h ∈ (0, − m 2 4l ) , one can obtain a class of periodic orbits of the dynamical structure Eq. ( 12).In this scenario, the Hamiltonian system will be written in the next formation . By employing Eq. ( 16) into the first equation of the Hamiltonian structure Eq. ( 12), and integrating we arrive with integral constant θ 0 .
Therefore, we obtain the next two periodic wave outcomes as For h = − m 2 4l , we have φ 2 1 = φ 2 2 = − m l , and the next kink wave (for positive sign) and antikink wave (for negative sign) solutions are obtained Case 2: l > 0, and m < 0 For h ∈ (− m 2 4l , 0) , one can acquire two classes of periodic orbits of the dynamical structure Eq. ( 12).In this case, the Hamiltonian system will be written in the subsequent formation . By employing Eq. ( 18) into the first equation of the Hamiltonian structure Eq. ( 12), and integrating we arrive and then we acquired the next two periodic wave outcomes as For h = 0 , we have φ 2 = 0 and φ 1 = − 2m l , and the next two bright (for positive sign) and dark (for negative sign) bell solitary wave outcomes are obtained For h ∈ (0, +∞) , the Hamiltonian system will be written in the following way . By employing Eq. ( 21) into the first equation of the Hamiltonian structure Eq. ( 12), and integrating we arrive ( 16)

Discussion of results
Using appropriate parameter values, we describe numerical simulations of the obtained outcomes and give their physical interpretation.For solution q 4 , the positive sign is displayed with a bright soliton, while the negative sign is displayed with a dark soliton.It is shown in Fig. 6 how the physical nature of the precise outcome q 4 will change when θ 0 = 1, α = 0.5, a 1 = a 2 = 1, a 3 = 1 2 , and a 4 = 2 .It is observed that for the positive sign, Fig. 6a,b is signified by a bright soliton, while for the negative sign, Fig. 6c,d is signified by a dark soliton.To examine the physical properties of the precise outcome, q 2 , a numerical simulation is presented in Fig. 7 for the parameters θ 0 = 1, α = 0.8, a 1 = 2, a 2 = −1, a 3 = a 4 = 1, and h = 1 8 .One can observe from the figure that for the positive sign, Fig. 7a,b is signified by a kink wave, while for the negative sign, Fig. 7c,d is signified by an anti-kink wave.Solutions q 1 , q 3 , and q 5 exhibit a periodic wave.Finally, to examine the physical properties of the precise outcome, q 1 , a numerical simulation is presented in Fig. 8 for the parameters θ 0 = 1, α = 0.5, a 1 = 2, a 2 = −1, a 3 = a 4 = 1, and h = 1 10 .

Stability analysis
To analyze the stability of the WBBM equation, this research will conduct linear stability analysis, as described in 38 .Assume WBBM's integer order, as stated in Eq. ( 4).Now, the perturbation solution takes the following structure: with incident power r.Inserting equation Eq. ( 23) into Eq.( 4), one reaches (23) q(x, y, z, t) = r + s(x, y, z, t),  The examination delves into the analysis of propagation relationships Eq. ( 28) as described in Fig. 9. Figure shows that the positive sign of ω(c, d, e) indicates whether the solution will amplify or diminish over time.The incident power r and wave numbers c, d, and e are stable states for small perturbations (red and green curves).
The system remains marginally stable when ω(c, d, e) = 0 , as disturbances do not grow or decay over time (yellow curve).Additionally, ω(c, d, e) is negative, the system moves further away from equilibrium over time, leading to instability in steady-state solutions and exponential distortion growth (blue and purple curves).

Novelty of the outcomes
This paragraph compares our results with recent studies, illustrating our findings' novelty.A literature review cited in [56][57][58][59] is included to determine the originality of our results.Mamun and his collaborators presented solitary and periodic wave solutions of the suggested model taking advantage of the (G ′ /G 2 )-expansion process 56 .Akram and his coauthors obtained some traveling wave solutions to this model employing the EMAEM method 57 .Inc and his colleagues solved this model through the Sarder-subequation scheme 58 .Kaabar and others investigated the suggested model utilizing the generalized Kudryashov process and exp(−φ(ζ )) technique 59 .Our results q 1 , q 2 , q 3 , q 4 , and q 5 exhibit novelty, when compared to their corresponding solutions.Notably, stability, bifur- cation, chaos, and sensitivity analysis of the governing model have not been reported in the literature.Consequently, our comparison highlights the novelty of additional solutions, representing the first instances of their construction for the investigated model.

Conclusion
We have effectively investigated bifurcation analysis and new waveforms to the first fractional 3D WBBM equation that appeared in shallow water waves.Moreover, the linear stability process is performed to assess the model's stability.By implementing the Galilean transformation, we have successfully obtained the dynamical system of (24) (s y s 2 2 + 2s y s r + s y r 2 + s t + s x − s zxt ) = 0, the mentioned model, facilitating a comprehensive bifurcation analysis.Additionally, we explored various solitary wave solutions, including dark solitons, bright solitons, kink waves, periodic waves, and anti-kink waves.
Through simulations, we visually presented these solutions, emphasizing their distinct characteristics and existence.The results highlight the effectiveness, brevity, and efficiency of the employed integration methods.They also suggest their applicability to delving into more intricate nonlinear models emerging in modern science and engineering scenarios.